Grades 5-6 Video Solutions 2022-2022_5-6

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Question number 20. Jesse writes the 7 numbers 3, 4, 5, 6, 7, 8, and 9 in the circles in the picture so that the sums of the 3 numbers on each line are equal.  What is the largest possible sum of 3 numbers on a line that Jesse can get? The first thing we want to find is the total sum. The total sum is 42.  From there, what we know is that one of these 7 numbers is in the middle. We'll call that x. The other 6 numbers are split into 3 pairs. The pair that goes here, here, and here.  Each of these 3 pairs has an equal sum. How do we know this? We're given that the sums of the 3 numbers on each line are equal.  If we add to each one of these pairs the same number x, then they'll be equal.  In other words, if we take this line and then we subtract out what they all have in common, we'll still have the same number.  The common sum of the pairs is the total sum of all numbers minus the middle divided by 3.  That gives us this value of this pair over here, this pair over here, this pair over here. They're all the same.  What's the sum of all 3 numbers on the line? That would be the sum of the pair plus the middle. 42 minus x over 3 plus x. We can rewrite this as 3x over 3.  We get in the numerator 42 minus x plus 3x all over 3.  Negative x plus 3x gives us 2x. We end up with this value of 42 plus 2x over 3. The maximum value of x is 9. The maximum sum is 42 plus 2 times 9 over 3.  42 plus 18 is 60. 60 divided by 3 is 20. Because of that, the maximum value is 20e.

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